The first $R_{cr}$ as a possible measure of the entrainment length in a 2D steady wake

Abstract

At a fixed distance from the body which creates the wake, entrainment is only seen to increase with the Reynolds number ($R$) up to a distance of almost 20 body scales. This increase levels up to a Reynolds number close to the critical value for the onset of the first instability. The entrainment is observed to be almost extinguished at a distance which is nearly the same for all the steady wakes within the $R$ range here considered, i.e. [20-100], which indicates that supercritical steady wakes have the same entrainment length as the subcritical ones. It is observed that this distance is equal to a number of body lengths that is equal to the value of the critical Reynolds number ($\sim 47$), as indicated by a large compilation of experimental results. {\it A fortiori} of these findings, we propose to interpret the unsteady bifurcation as a process that allows a smooth increase-redistribution of the entrainment along the wake according to the weight of the convection over the diffusion. The entrainment variation along the steady wake has been determined using a matched asymptotic expansion of the Navier-Stokes velocity field [Tordella and Belan, {\it Physics of Fluids}, 15(2003)] built on criteria that include the matching of the transversal velocity produced by the entrainment process.